Introduction to Differential Equations
Part 3: Slope fields
We have examined a number of first-order differential equations of the form
dY/dt = f(t,Y)
For example, the differential equation dY/dt = t - Y is of this form with f(t,Y) = t - Y. Sometimes, either the independent variable or the dependent variable is not present in the formula for f. An example of the first case is the differential equation modeling the spread of the rumor,
dS/dt = k S (M - S).
Here, the dependent variable is S, and the independent variable t is missing from the right-hand side of the equation.
An example of the case where the dependent variable is missing from the right-hand side is dY/dt = t^{2} + 1. The point is that both of these cases are included in the class we are discussing in this part.
So suppose we are considering a differential equation of the form
dY/dt = f(t,Y).
Can we determine what the graphs of solutions to this differential equation look like without obtaining symbolic descriptions of the solutions? What we can determine easily is the slope of the graph of a solution as it passes through the point (t_{0}, y_{0}). That slope is just f(t,Y) -- but how does that help us?
To see how, let's consider the equation
dY/dt = t - Y.
Imagine the plane to be completely covered by graphs of solutions to this differential equation.
- On your worksheet draw in the graphs of six different solutions of this equation.
In the figure below, we have drawn (in green) the graph of the solution that passes though (1,3), i.e., the solution satisfying the initial condition Y(1) = 3. In addition to the graph (in green), we have drawn a short piece (in blue) of the tangent line to the graph at (1,3).
- What is the slope of the pictured tangent line?
In a "small" region about the point (1,3), the tangent line is a reasonable approximation to the graph of the solution.
- Let t be a point near 3, and let (t,y) be the corresponding point on the tangent line. Give two factors that determine how well y approximates Y(t).
So if we have enough tangent line segments, we have a good idea what the graphs of the solutions look like -- and we can sketch the tangent lines just knowing the right-hand side of the differential equation. A grid of these short tangent line segments is called a slope field or direction field. Here is a slope field for the equation dY/dt = t - Y.
Looking at this slope field, you should be able to imagine a variety of solution graphs.
- Pick a point in the second quadrant, say (-3,3). Describe in words what the graph of the solution through this point should look like.
- What are the slopes along the line Y = t? Determine these slopes from the differential equation itself, and then confirm this from the slope field.
- What are the slopes along the horizontal axis, Y = 0?
- What are the slopes along the vertical axis, t = 0?
- What slopes do you find along the line Y = t- 1? Explain why the function Y = t- 1 is a solution of the differential equation.
- Is there any other linear function that is a solution? Explain your answer.
- Describe in words the solution functions (or the graphs of solution functions) other than the linear function in Step 8. In particular, what happens to solution functions as t becomes large?
- Your worksheet has been set up to draw a slope field like the one above,
and to add three solution curves. Carry out the steps in the worksheet, and make sure the results confirm your answers to the questions above.
Your answers to the questions above demonstrate that you can get a lot of information about solutions to a differential equation directly from the slope field -- a picture of the problem -- even without calculating any algebraic forms for solutions. In fact, we even found an algebraic formula for one particular solution -- one that turned out to be important for describing all the others.
- Draw the slope field for the differential equation we used to model the spread of a rumor
dS/dt = k S (M - S).
Recall that M is the population of your school and that we chose k = 0.00025. Experiment with the range values for the independent and dependent variables until your slope field gives good information on the graphs of solutions. What geometric feature of the slope field reflects the fact that the slope depends only on S, not on t?
- Describe geometric features of the slope field that reflect symbolic (i.e., algebraic) features of the slope function.
- Plot the slope field along with two "solution curves," i.e., graphs of solutions, with distinctly different shapes. You may need to experiment with various initial conditions to obtain appropriate solution curves.
- In Part 2 of the worksheet, you were given the solution to the initial value problem with S(0) = 2, k = 0.00025 and M = population of your school.
Plot that particular function, and superimpose its graph over your slope field. Does it look like it fits?
- The same differential equation we used to model the spread of a rumor also can be used to model the growth of an animal population in an environment that limits the number of individuals that can be supported. In this context, M is called the "carrying capacity" of the species. Describe what this model predicts if the initial population is greater than the carrying capacity. Explain this result algebraically using the differential equation.
- For each of the differential equations given below, do the following:
- Draw a slope field for the differential equation. (Experiment with the range values.)
- Write a brief description of the general behavior of the solution curves for the
equation.
- Add three solution curves to the slope field which illustrate your description.